The Cure For Graph Dyslexia (And It’s Not A Pretty Sight)

I don’t want to count any chickens before they hatch, but I do believe I may have cured my “reading functions backwards under pressure” problem. Incidentally, my son told me lots of people have this issue, which made me feel better.

It took me a few tries, and I even double stumped myself — but here is the most gruesome Function Notation Graph question I could conjure up:

The figure above shows the graphs of the functions f and g.  If  y = f(x), and f(4) = k,  and g(-2) = m, which of the following is closest to f(k) + g(m) - (k/m)² ?

(A)  f(-2)

(B)  f(2)

(C)  g(-3)

(D) g(-6)

(E)  g(6)

What do you think?  All answers/attempts/questions left in the comments will Make My Day.

By the way, is anyone else having a hard time finding their SAT groove? I feel like there is always something “really important” to do before I can get to my SAT work, and then I never seem to get there (“there” = SAT work).

I’m scheduled to take a full, timed PSAT with my son tomorrow morning — so hopefully that will be my “get back on the horse” moment.

Illustrations by Jennifer Orkin Lewis

#PerfectScoreProject

Associative Interference

A few months ago I joked, How Long Till the Polynomials?

All kidding aside, I knew there was a pebble in my shoe around the polynomials, I just couldn’t pinpoint the issue at the time.

Turns out it was more than a pebble.

Cut to a few weeks ago, and I was attempting to write my own “Solve for Expressions” question. I emailed the first draft to PWNtheSAT so he could take it for a test drive, and I got back the following message:

“I don’t see how to get from what you’re giving me to what you want……Is there a trick I’m not seeing?”

That was my first inkling that something was very wrong, but I checked my work and sent him my steps:

First this: (a - b)², then that: (a + b)(a - b), etc. etc.

And then he emailed me again:

“Look at your second step!”

And just like that, in the blink of an eye, I had my polynomial epiphany.

(Incidentally, I’m baring my soul here in case there’s anyone out there who might benefit from knowing that it’s okay not to know everything.)

Mortified, I wrote back, “I’m scaring you now, right? I’m beyond your scope, aren’t I?”

Then he told me it’s a big distinction, but a common mistake (and I am choosing to believe him about the “common mistake” part, if only to maintain the courage to soldier on, and not die from embarrassment.) And, I’ll try not to obsess about whatother holes might be lurking.

I called my friend Catherine who attempted to console me. “It’s not you,” she said, “It’s called associative interference. Have you read Wickelgren?”

And then she sent me a post she’d written, from which I will quote, because it didmake me feel better: Why is Remembering What You’ve Learned About Math Hard?

It’s the similarity between the facts. That is, the fact 3 + 5 = 8 is not so different from 3 + 6 = 9. They both contain 3’s; they both contain +’s, and they both contain single-digit numbers….

Thus, to a child beginning to learn such facts, the facts overlap in the brain, creating a blur that makes it easy to confuse them and difficult to remember any single answer. In cognitive psychology, this “blur” is called associative interference, which occurs when one idea, A, is linked in the mind to two or more other ideas. It’s like static on the radio, which often occurs when other stations or electrical impulses interfere with a radio station’s music or speech.

Anyhoo, I adapted my “Solve for Expressions” question to incorporate all areas of confusion:

If a² = 4 and b² = 9, which of the following could equal c in the following equation: c(a - b)² = 2(a² - b²)

A) -10

B) -2

C) .5

D) 2

E) 25

Hopefully, “the issue” is now resolved. I did have a moment of satisfaction when I ran across a need to know this piece of information yesterday, while taking a full, timed, practice test.

As always, any and all attempts to answer the question above in the comments below, will make my day.

Illustrations by Jennifer Orkin Lewis

Organizing the Mental Closet

Catherine and I have been going back and forth about “associative interference,” since coincidentally we simultaneously wrote blog posts about the issue.

We’ve decided that not knowing the proper math and grammar terminology adds a layer of difficulty to the process of trying to improve.

For example, try getting to the bottom of your errors with explanations like these (from the College Board):

It avoids the comma-splice error of the other options by turning the first independent clause, “This basic document is stating the liberties” into an appositive. An appositive is asubordinate noun phrase that renames a noun. In this revision, “A basic document” is the appositive that renames “Magna Carta,” and the dependent clause “that states the liberties” modifies “a basic document.”

Woof woof woof.

I don’t get it.

Ok, I do finally know how to identify a “comma splice” (now, age 45, and 10 months into studying for the SAT), but I’m not sure I could pick out a “dependent clause” from an “independent one,” and I would need the Google machine to help me understand the ”appositive” portion of this explanation.

How am I supposed to organize my mental closet if I don’t know the jargon?

According to Catherine, the terminology — say “gerund” or “dangling participle” or “noun clause, ” keeps reminding you that these are “different grammatical structures.”

“Knowledge isn’t just facts and notions,” she explained to me, “It’s facts and notions etc. organized inside a SCHEMA.”

From Catherine’s post about associative interference:

I’m also thinking more attention should be paid to teaching young children the terminology of arithmetic: addends, subtrahends, factors, and the like. I think — I don’t know — that fluency with the terminology might help reduce associative interference. “All math looks alike”: the 5 and the 2 in 5+2 look exactly like the 5 and the 2 in 5x2. But the words addend and factor have nothing in common whatsoever.

Illustrations by Jennifer Orkin Lewis

Watch Your Back (That’s All I’m Sayin’)

Or “Savor the Flavor,” as PWNtheSAT said to me yesterday, as I screamed with rage after being messed with, again (and again and again), by the SAT math section.

I inadvertently struck a nerve a week or two ago when I suggested that you need to know more than just solid math to do well on the math portion of the SAT.

One (anonymous) commenter even left me this message on Psychology Today:

“This idea that the test is full of booby traps is ridiculous. You simply have to READ the question, figure out what they are asking, and then answer accordingly. You need to show that you understand the basic math concepts. The questions aren’t tricks. I had read a version of that assessment over and over again, and then took the test again as an adult and easily scored 800. BECAUSE I UNDERSTAND MATH AND CAN READ.”

But I’m sticking with my position: Solid math basics are essential, but not sufficient. If you want to ace this test, you need to be prepared not to be messed with by a test that’s trying to mess with you.

Take, for example, the following questions, all of which come from just one, lone, College Board practice SAT, and I believe illustrate the point that “solid” math knowledge alone is not enough (at least not while the clock is ticking you’ve got about one minute per problem).

Exhibit 1:  

This appeared (to me) to be run of the mill parabola question, so I broke the “Quadratic Equation” and jotted it down in the margins: y = ax² + bx + c.  Then, I did my very best to turn what they gave me, back into what I knew.

But I couldn’t get past that minus sign in between the “a” and the “x².” No idea what that meant.

Well I’ll tell you what it meant:

It meant that the “a” referred to in this problem (above) is not the same ”a” that I learned about in “math” — It’s just coincidentally also called “a” — just like the one that’s usually located in the exact same position.  

But don’t be confused. It’s not THAT ”a.”

MEAN….mean mean mean. Makes me scream.

Exhibit 2:

Looked like a 30 60 90 to me.

Wrong!  ”Not drawn to scale” = dead giveaway.

And don’t let those (a + b)² send you down the “Pythagorean road” that you learned in math class, because it’s not that either (go figure).

This, my friends, would be the fanciest “Sadness Gap” question I ever did see.

(Who would have thunk. Not me, that is for sure.)

Exhibit 3:

So (so so so so) proud of myself, and brimming with enthusiasm at the prospect of trying out my newfound Polynomial clarification, I FOILed the thing.

In fact, I probably spent a good 2 minutes down FOIL ROAD — never arriving at the answer until after the bell, when I went back and looked, and went “dah, I can’t believe I did that.”

They got me again.

Illustrations by Jennifer Orkin Lewis

 

I’m Feeling Icarus

Why did I have to effuse in such an absurdly over the top manner the other day?

I’m pretty sure that’s where my bad luck began.

Trust me when I say that the jubilation came to a screeching halt right after I tallied my scores from a full practice SAT the very next morning.

Not a pretty sight.  And humbling.

Please stop me if you ever see me doing that happy dance again.

This leads to a pile of melted feathers.

Anyway, 99.99% of the people who emailed after score day were of the extremely“supportive” variety (which made me feel so good.  So thank you).

“You’re an inspiration to know that this can be conquered with some motivation and persistence.”

“Congrats!!!!! 800 in verbal is awesome!! if only the SAT gods would be so merciful to me :O) “

“You are totally cracking me up! And making me a little less afraid to take the GRE.” 

 “OMG perfectscoreproject…..you’re the coolest mom ever.”

Even my own teenage daughter told me how proud she was of me.

But of course there had to be one email (from a tutor offering advice) that I never should have opened up right after scoring that 5 hour (punishing) practice SAT. And on a belly full of nothing more than a few chocolate fumes, I began to read this lengthy email:

She told me that I’m on the wrong track, that this test is “ridiculously easy,” and that the kids at the top schools (“including her younger self”) don’t want to be in classes with kids who can’t answer these questions.  ”They are so basic,” she said.  And then she added that the fact that I haven’t been given a good math education shows up in my score, “and my writing.”

Ow.

I perseverated for days. (And yes, I use this word a lot. I like it.)

And then I woke up this morning and thought to myself, you know, I’m standing by my opinion: This test is hard.  

Say what you will, but I urge you to give it a go yourself if you’ve got a kid coming up to bat in the next few years.  The College Board offers a free practice SAT on their website.  Take it all at once, and timed, so you can experience the full effect.

My friend Catherine, at Kitchen Table Math, has written a few posts recently about the difficulty of the SAT that are well worth the read.

One father wrote to me that his daughter, a high school junior, seemed to pick the math up very easily.  After a few weeks he made a judgement call not to spend their limited resources (i.e. time) on the math section, but rather focus on the reading and writing instead:

“Where most of the solutions to the SAT questions are rather simple and straight forward if you can get the “trick” to the question.  I mention all this because the math is somewhat of a gift in that the ones who have math “insight” can see the trick and quickly answer the question.  And getting a physics degree at Columbia doesn’t necessary mean you have that gift. Part of what happened with my daughter is she started “seeing” the insights necessary to answer the questions.”

I suspect he is right.  There is a degree of “gift” and “insight” that is beyond the scope of how well educated you are and how hard you’ve worked.  And different people have different gifts.

Anyway, enough about this from me for now.

Charts & Graphs have been updated to reflect the latest scores.

Illustrations by Jennifer Orkin Lewis